MinT - Best Known (227−111, 227, s)-Nets in Base 3

Best Known (227−111, 227, s)-Nets in Base 3

(227−111, 227, 76)-Net over F₃ — Constructive and digital

Digital (116, 227, 76)-net over F₃, using

1 times m-reduction [i] based on digital (116, 228, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 71, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 157, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(227−111, 227, 120)-Net over F₃ — Digital

Digital (116, 227, 120)-net over F₃, using

t-expansion [i] based on digital (113, 227, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(227−111, 227, 920)-Net in Base 3 — Upper bound on s

There is no (116, 227, 921)-net in base 3, because

1 times m-reduction [i] would yield (116, 226, 921)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 676477 846269 673730 630681 464804 168281 844032 399730 165139 820457 124095 518430 183032 934151 878798 306336 729045 284651 > 3²²⁶ [i]